A Systematic Method for the Analytical Evaluation of Convolution Integrals
Irwin S. Goldberg, Michael G. Block, and Roland E. Rojas
Abstract—Analytic methods are presented for the systematic evaluation of convolution integrals that contain piecewise-continuous or piecewise-smooth functions. These methods involve breaking theconvolution integral into a sum of integrals and expressing each integral in a standard form that can be easily evaluated. Mathematical insight is gained by looking at the result as a superposition of terms, where each term is nonzero or active during a specific time interval. Applications to both causal and noncausal linear systems are considered. Additionally, an example is included thatprovides students with insight into the construction and the behavior of the impulse function. These methods have been successfully taught to undergraduate engineering students to enhance their understanding of convolution and to facilitate their ability to evaluate convolution integrals. Index Terms—Convolution, convolution integral, impulse function, impulse response.
ELECTRICAL engineering students, during their junior year at St. Mary’s University, San Antonio, TX, are required to take a Linear Signals and Systems Analysis course. In this course, the students are introduced to continuous-time convolution integrals. For a continuous, linear, time-invariant system with an im, the response function, , to an input pulse response function, , is determined by aconvolution integral. This , is convolution relation, denoted as (1a) where is a dummy integration variable. Alternatively, because can also be evaluated as (1b) The students are first taught to calculate continuous-time convolution results by evaluation of the above superposition integral in the time domain. Later, the students are taught to use transform methods to calculate continuous-time convolutionresults. Semigraphical methods are commonly presented in textbooks for the time-domain evaluation of the convolution integral –. Students often find these semigraphical
methods for the evaluation of the integrals to be difficult and error prone, especially when discontinuous functions are involved. In contrast, a systematic analytical method will be presented for the time-domain evaluationof convolution integrals containing piecewise-continuous functions and/or piecewise-smooth functions. This analytical method involves the following steps: and/or Step 1) expressing each discontinuous function, , as sums and differences of terms containing unit-step functions; Step 2) substituting the dummy integration variable for the independent variable in one function, either or . Also,substituting for the independent variable in the other function; Step 3) writing the integral of the product [or of the product ] as a sum of integrals—each integral containing a unit-step function or a product of two unit-step functions; Step 4) evaluating each integral in Step 3) in terms of standard integral forms given in (4b), (23), (24b), and (25). The result is expressed in terms of sums ofdelayed unit-step functions. The following three properties are useful as checks of the convolution result. 1) When both of the two convoluted functions are bounded (i.e., when impulse functions are not included), the result of the convolution integral is a continuous function of . Particularly, the result should be continuous at points where the functional form of the solution abruptly changes. 2) Thearea of the convolution result is equal to the product of the areas of the two functions that are involved in the convolution operation –. 3) When the two convoluted functions are of finite time duration (that is, when both functions are of bounded support), the total duration of the result is equal to the sum of the durations of the two functions involved in the convolution operation ,...